Graded rings and essential ideals (original) (raw)
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Let G be a group and R be a G-graded commutative ring, i.e., R = ⊕ g∈G Rg and Rg Rh ⊆Rgh for all g, h ∈G. In this paper, we study the graded primary ideals and graded primary G-decomposition of a graded ideal.
On Generalizations of Graded rrr-ideals
2021
In this article, we introduce a generalization of the concept of graded r-ideals in graded commutative rings with nonzero unity. Let G be a group, R be a G-graded commutative ring with nonzero unity and GI(R) be the set of all graded ideals of R. Suppose that φ : GI(R) → GI(R) ⋃ {∅} is a function. A proper graded ideal P of R is called a graded φ-r-ideal of R if whenever x, y are homogeneous elements of R such that xy ∈ P − φ(P ) and Ann(x) = {0}, then y ∈ P . Several properties of graded φ-r-ideals have been examined.
GR-N-Ideals in Graded Commutative Rings
Acta Universitatis Sapientiae, Mathematica, 2019
Let G be a group with identity e and let R be a G-graded ring. In this paper, we introduce and study the concept of gr-n-ideals of R. We obtain many results concerning gr-n-ideals. Some characterizations of gr-n-ideals and their homogeneous components are given.
Commutativity and Ideals in Strongly Graded Rings
Acta Applicandae Mathematicae, 2009
In some recent papers by the first two authors it was shown that for any algebraic crossed product A, where A 0 , the subring in the degree zero component of the grading, is a commutative ring, each non-zero two-sided ideal in A has a non-zero intersection with the commutant C A (A 0 ) of A 0 in A. This result has also been generalized to crystalline graded rings; a more general class of graded rings to which algebraic crossed products belong. In this paper we generalize this result in another direction, namely to strongly graded rings (in some literature referred to as generalized crossed products) where the subring A 0 , the degree zero component of the grading, is a commutative ring. We also give a description of the intersection between arbitrary ideals and commutants to bigger subrings than A 0 , and this is done both for strongly graded rings and crystalline graded rings.
Some notes on first strongly graded rings
Miskolc Mathematical Notes, 2017
Let G be a group with identity e and R be an associative ring with a nonzero unity 1. Assume that R is first strongly G-graded and H D supp.R; G/. For g 2 H , define˛g .x/ D n g X i D1 r .i / g xt .i / g 1 where x 2 C R .R e / D fr 2 R W rx D xr for all x 2 R e g, r .i / g 2 R g and t .i / g 1 2 R g 1 for all i D 1; :::::; n g for some positive integer n g. In this article, we study˛g .x/ and it's properties.
Journal of the Mathematical Society of Japan, 1978
In this paper, we study a Noetherian graded ring RRR and the category of graded R-modules. We consider injective objects of this category and we define the graded Cousin complex of a graded R-module MMM . These concepts are essential in this paper
Associated graded rings of one-dimensional analytically irreducible rings II
Journal of Algebra, 2011
Lance Bryant noticed in his thesis [3], that there was a flaw in our paper [2]. It can be fixed by adding a condition, called the BF condition in [3]. We discuss some equivalent conditions, and show that they are fulfilled for some classes of rings, in particular for our motivating example of semigroup rings. Furthermore we discuss the connection to a similar result, stated in more generality, by Cortadella-Zarzuela in [4]. Finally we use our result to conclude when a semigroup ring in embedding dimension at most three has an associated graded which is a complete intersection. 2000 Mathematics Subject Classification: 13A30 If x ∈ R is an element of smallest positive value, i.e. v(x) = e, then xR is a minimal reduction of the maximal ideal, i.e. m n+1 = xm n , for n >> 0. Conversely each minimal reduction of the maximal ideal is a principal ideal generated by an element x of value e. The smallest integer n such that m n+1 = xm n is called the reduction number and we denote it by r. Observe that, if v(x) = e, then Ap e (S) = S \(e+S) = v(R)\v(xR), therefore w j / ∈ v(xR), for j = 0,. .. , e − 1. Consider the m-adic filtration m ⊃ m 2 ⊃ m 3 ⊃. .. . If a ∈ R, we set ord(a) := max{i | a ∈ m i }. If s ∈ S, we consider the semigroup filtration v(m) ⊃ v(m 2) ⊃. .. and set vord(s) := max{i | s ∈ v(m i)}. If a ∈ m i , then v(a) ∈ v(m i) and so ord(a) ≤ vord(v(a)). According to [3], we say that the m-adic filtration is essentially divisible with respect to the minimal reduction xR if, whenever u ∈ v(xR), then there is an a ∈ xR with v(a) = u and ord(a) = vord(u). The m-adic filtration is essentially divisible if there exists a minimal reduction xR such that it is essentially divisible with respect to xR. We fix for all the paper the following notation. Set, for j = 0,. .. , e − 1, b j = max{i|w j ∈ v(m i)}, and let c j = max{i|w j ∈ v(m i + xR)}. Note that the numbers b j 's do not depend on the minimal reduction xR, on the contrary the c j 's depend on xR. Lemma 1.1 If I and J are ideals of R, then v(I +J) = v(I)∪v(J) is equivalent to v(I ∩ J) = v(I) ∩ v(J).
On Graded Semiprime and Graded Weakly Semiprime Ideals
2013
Let G be an arbitrary group with identity e and let R be a Ggraded ring. In this paper, we define graded semiprime ideals of a commutative G-graded ring with nonzero identity and we give a number of results concerning such ideals. Also, we extend some results of graded semiprime ideals to graded weakly semiprime ideals. Mathematics Subject Classification (2010):13A02, 13C05, 13A15
Associated graded rings of one-dimensional
2010
Lance Bryant noticed in his thesis [3], that there was a flaw in our paper [2]. It can be fixed by adding a condition, called the BF condition in [3]. We discuss some equivalent conditions, and show that they are fulfilled for some classes of rings, in particular for our motivating example of semigroup rings. Furthermore we discuss the connection to a similar result, stated in more generality, by Cortadella-Zarzuela in [4]. Finally we use our result to conclude when a semigroup ring in embedding dimension at most three has an associated graded which is a complete intersection. 2000 Mathematics Subject Classification: 13A30 If x ∈ R is an element of smallest positive value, i.e. v(x) = e, then xR is a minimal reduction of the maximal ideal, i.e. m n+1 = xm n , for n >> 0. Conversely each minimal reduction of the maximal ideal is a principal ideal generated by an element x of value e. The smallest integer n such that m n+1 = xm n is called the reduction number and we denote it by r. Observe that, if v(x) = e, then Ap e (S) = S \(e+S) = v(R)\v(xR), therefore w j / ∈ v(xR), for j = 0,. .. , e − 1. Consider the m-adic filtration m ⊃ m 2 ⊃ m 3 ⊃. .. . If a ∈ R, we set ord(a) := max{i | a ∈ m i }. If s ∈ S, we consider the semigroup filtration v(m) ⊃ v(m 2) ⊃. .. and set vord(s) := max{i | s ∈ v(m i)}. If a ∈ m i , then v(a) ∈ v(m i) and so ord(a) ≤ vord(v(a)). According to [3], we say that the m-adic filtration is essentially divisible with respect to the minimal reduction xR if, whenever u ∈ v(xR), then there is an a ∈ xR with v(a) = u and ord(a) = vord(u). The m-adic filtration is essentially divisible if there exists a minimal reduction xR such that it is essentially divisible with respect to xR. We fix for all the paper the following notation. Set, for j = 0,. .. , e − 1, b j = max{i|w j ∈ v(m i)}, and let c j = max{i|w j ∈ v(m i + xR)}. Note that the numbers b j 's do not depend on the minimal reduction xR, on the contrary the c j 's depend on xR. Lemma 1.1 If I and J are ideals of R, then v(I +J) = v(I)∪v(J) is equivalent to v(I ∩ J) = v(I) ∩ v(J).
Some properties of gr-multiplication ideals
Turkish Journal of Mathematics, 2009
In this paper, we study some of the properties of gr-multiplication ideals in a graded ring R . We first characterize finitely generated gr-multiplication ideals and then give a characterization of gr-multiplication ideals by using the gr-localization of R . Finally we determine the set of gr-P -primary ideals of R when P is a gr-multiplication gr-prime ideal of R .